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Mirrors > Home > MPE Home > Th. List > 2moswapv | Structured version Visualization version GIF version |
Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2645 with a disjoint variable condition, which does not require ax-13 2371. (Contributed by NM, 10-Apr-2004.) (Revised by Gino Giotto, 22-Aug-2023.) Factor out common proof lines with moexexvw 2629. (Revised by Wolf Lammen, 2-Oct-2023.) |
Ref | Expression |
---|---|
2moswapv | ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nfe1 2151 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
2 | 1 | nfmov 2559 | . . . 4 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
3 | nfe1 2151 | . . . . 5 ⊢ Ⅎ𝑥∃𝑥(∃𝑦𝜑 ∧ 𝜑) | |
4 | 3 | nfmov 2559 | . . . 4 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑) |
5 | 1, 2, 4 | moexexlem 2627 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
6 | 5 | expcom 417 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
7 | 19.8a 2178 | . . . . 5 ⊢ (𝜑 → ∃𝑦𝜑) | |
8 | 7 | pm4.71ri 564 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
9 | 8 | exbii 1855 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
10 | 9 | mobii 2547 | . 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
11 | 6, 10 | syl6ibr 255 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∀wal 1541 ∃wex 1787 ∃*wmo 2537 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-10 2141 ax-11 2158 ax-12 2175 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-tru 1546 df-ex 1788 df-nf 1792 df-mo 2539 |
This theorem is referenced by: 2euswapv 2631 2rmoswap 3671 |
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